After BENG in EENG

Hi,[My Problem]
I got a bachelors in Electrical Engineering about 2 years ago and since then I've just been floating around like an unguided wave. A major problem of mine is deciding what to pursue next. I am undoubtedly going to continue my education but not very sure what is exactly I should do. This might be surprising but the the reason for my hesitation is that I am interested in so many things that I'm overwhelmed. But lately I've been narrowing down my options but I'm still facing difficulties because I don't know what exactly such a course in called, hence this thread.

[What I want]
So, I'm very interested in electronics (micro) and I'd like to put myself in the position where I can do cutting edge research on the fabrication of modern electronic devices. I'm thinking of thinks like nanowires, nanotubes, modern semiconductor materials etc. I'm aware that such a course might require quantum mechanics and material science. With the exception of a usual freshman physics introduction into quantum mechanics, my EENG curriculum included neither QM nor MS.

[What to do]
Given that I have a bachelors in EENG and the above:
1. What courses fit my description ? (I mean what is the name of the graduate program I should be looking for)
2. Will I be eligible for such a course (because it seems very Physics based) ?
3. Since I am not verse in QM or MS, will such courses provide preliminary classes for students from different backgrounds such as EENG ?

I hope I have clearly explained what I'm looking for and I really hope I can get the help that I need to focus on the next stage of my carrier.